Reference Sheet · SAT Math · Geometry
Areas & Volumes
Area formulas for plane figures, volume formulas for solids, and the angle properties of polygons. The SAT gives you most of these in the reference box, but you should know them cold.
Plane figures (areas)
Rectangle
$$A = \ell w$$
ℓ = length
w = width
w = width
Length times width. Perimeter is $P = 2\ell + 2w$.
EXAMPLE
Rectangle: length $8$, width $5$. Find area.
$A = 8 \cdot 5 = $ $40$
Triangle
$$A = \frac{1}{2}bh$$
b = base
h = height (perpendicular to the base)
h = height (perpendicular to the base)
Height must be perpendicular to the base — not the slanted side. Works for any triangle, not just right triangles.
EXAMPLE
Triangle with base $10$, height $6$. Find area.
$A = \tfrac{1}{2}(10)(6) = $ $30$
Parallelogram
$$A = bh$$
b = base
h = height (perpendicular distance between the two parallel sides)
h = height (perpendicular distance between the two parallel sides)
Like a slanted rectangle. Don't use the slanted side as height — use the perpendicular distance.
EXAMPLE
Parallelogram: base $7$, height $4$. Find area.
$A = 7 \cdot 4 = $ $28$
Trapezoid
$$A = \frac{1}{2}(b_1 + b_2)h$$
b₁, b₂ = the two parallel sides
h = perpendicular distance between them
h = perpendicular distance between them
Average the two parallel sides, then multiply by the height.
EXAMPLE
Trapezoid: parallel sides $6$ and $10$, height $4$. Find area.
$A = \tfrac{1}{2}(6 + 10)(4) = \tfrac{1}{2}(16)(4) = $ $32$
Polygon angles
Sum of interior angles
$$\text{sum} = 180°(n - 2)$$
n = number of sides
Triangle ($n=3$): $180°$. Quadrilateral ($n=4$): $360°$. Pentagon ($n=5$): $540°$. Hexagon ($n=6$): $720°$.
EXAMPLE
Find the sum of interior angles of an octagon ($n = 8$).
$180°(8 - 2) = 180°(6) = $ $1080°$
Sum of exterior angles
$$\text{sum} = 360°$$
Always $360°$, regardless of how many sides. One of the most useful — and most forgotten — facts on the SAT.
EXAMPLE
Find each exterior angle of a regular hexagon.
$\dfrac{360°}{6} = $ $60°$
One interior angle of a regular polygon
$$\text{each angle} = \frac{180°(n - 2)}{n}$$
Total interior degrees divided by the number of angles. Only works for regular (equilateral and equiangular) polygons.
EXAMPLE
Find each interior angle of a regular pentagon.
$\dfrac{180°(5 - 2)}{5} = \dfrac{540°}{5} = $ $108°$
Triangle properties
Interior angle sum
$$\angle A + \angle B + \angle C = 180°$$
All three interior angles of any triangle add to $180°$.
EXAMPLE
Two angles of a triangle are $50°$ and $70°$. Find the third.
$180° - 50° - 70° = $ $60°$
Exterior angle theorem
$$\text{exterior} = \text{sum of remote interiors}$$
An exterior angle equals the sum of the two interior angles that aren't next to it.
EXAMPLE
Two interior angles are $40°$ and $65°$. Find the exterior angle at the third vertex.
$40° + 65° = $ $105°$
Pythagorean theorem
$$a^2 + b^2 = c^2$$
a, b = legs (the two shorter sides)
c = hypotenuse (across from the right angle)
c = hypotenuse (across from the right angle)
Only works for right triangles. The hypotenuse is always the longest side, opposite the right angle.
EXAMPLE
Right triangle with legs $6$ and $8$. Find the hypotenuse.
$c^2 = 36 + 64 = 100 \ \Rightarrow \ c = $ $10$
Pythagorean triples
$$3\text{-}4\text{-}5,\ \ 5\text{-}12\text{-}13,\ \ 8\text{-}15\text{-}17$$
Right triangles with all-integer sides. Memorize these — and their multiples (6-8-10, 9-12-15, etc.). Spotting them saves the work of computing the Pythagorean theorem.
EXAMPLE
Right triangle with legs $9$ and $12$. Find the hypotenuse.
Recognize $9\text{-}12\text{-}? = 3 \cdot (3\text{-}4\text{-}5)$, so hyp $= $ $15$
3D solids (volumes)
Rectangular prism (box)
$$V = \ell w h$$
ℓ = length, w = width, h = height
Multiply all three dimensions. A cube has $V = s^3$ where $s$ is the side length.
EXAMPLE
Box with dimensions $3 \times 4 \times 5$. Find volume.
$V = 3 \cdot 4 \cdot 5 = $ $60$
Cylinder
$$V = \pi r^2 h$$
r = radius of the circular base
h = height
h = height
Area of the circular base, times the height.
EXAMPLE
Cylinder with radius $3$, height $10$. Find volume.
$V = \pi(3)^2(10) = $ $90\pi$
Sphere
$$V = \frac{4}{3}\pi r^3$$
r = radius
Cube the radius first, then multiply by $\tfrac{4}{3}\pi$.
EXAMPLE
Sphere with radius $3$. Find volume.
$V = \tfrac{4}{3}\pi(27) = $ $36\pi$
Cone
$$V = \frac{1}{3}\pi r^2 h$$
r = radius of the circular base
h = height
h = height
Exactly $\tfrac{1}{3}$ of the cylinder with the same base and height.
EXAMPLE
Cone with radius $3$, height $10$. Find volume.
$V = \tfrac{1}{3}\pi(9)(10) = $ $30\pi$
Pyramid
$$V = \frac{1}{3} \cdot \text{base area} \cdot h$$
base area = area of the base (could be rectangle, triangle, etc.)
h = perpendicular height from base to apex
h = perpendicular height from base to apex
Same $\tfrac{1}{3}$ relationship as the cone — pyramid is to a prism what a cone is to a cylinder.
EXAMPLE
Pyramid with rectangular base $4 \times 6$ and height $5$. Find volume.
Base area $= 24$. So $V = \tfrac{1}{3}(24)(5) = $ $40$